The point of inflexion of a function \( f(x) \) is the point where:
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Show Hint
- \( f'(x) = 0 \) and \( f'(x) \) changes its sign from positive to negative from left to right of that point.
- \( f'(x) = 0 \) and \( f'(x) \) changes its sign from negative to positive from left to right of that point.
- \( f'(x) = 0 \) and \( f'(x) \) does not change its sign from left to right of that point.
- \( f'(x) \neq 0 \).
{5pt}
The Correct Option is C
Solution and Explanation
Step 1: Definition of a point of inflection.
A point of inflection is a point where the concavity of the function changes. This means \( f''(x) \) changes its sign at that point.
Step 2: Conditions for inflection points.
At a point \( x = c \), the function \( f(x) \) has a point of inflection if: - \( f''(x) \) changes its sign around \( x = c \). - \( f'(x) = 0 \) but does not change its sign.
Step 3: Analyzing the options.
Option (C) correctly states that \( f'(x) = 0 \) and \( f'(x) \) does not change its sign, which is consistent with the definition of a point of inflection.
Step 4: Conclusion.
The correct answer is (C).
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